170 XIII INTEGRATION converges. Note that if m2 g N < (m + I)2, we have , . m2 '<*)--/.ğ<> and deduce that the sequence (/N(X))NğI tends to 0 for almost all x e [0,1] (with respect to Lebesgue measure). Hence show that, for almost all x e [0,1], the sequence (An* [An;t]) is equirepartitioned with respect to Lebesgue measure (Section 13.4, Problem 7). 16. Let U be a continuous endomorphism of the space LR(X, p) with norm <^ 1 (in other words, such that NI(£/ /) <£ NI(/)) and such that the relation/^ 0 implies U-f^O (13.6). If/is any function in the class f, we denote by U /any function in the class u-f. (a) Let (fn) be a sequence of functions in Ğ^R which converges almost everywhere to a function/and is such that there exists a function h e 3?^ satisfying \fn\ ^ /z for all n. Show that under these conditions the sequence (U-fn) converges almost every- where to £/*/ (Reduce to the case /= 0, and consider the sequence of functions (b) For each finite sequence (tk)i;^k^n of real numbers, put , ...,/) >0, then S-n('l,...sO = SUP (*i + - l<fc<n Show that if sn(tlf . . . , tn) > 0, then for all /B+1 6 R we have (1) *i + sn(t2 , . . . , tn+1) ^ 5n(/i, . . . , tn). (c) For each function / e ^R , show that sJiU-f, . . . , £/n+1 /) < t/ - JH(/, ...,(/" /) almost everywhere. (Use (1) and the fact that for any n functions fl9 ...,/ 6 &^ we have sup U fk ^ U ' sup fk l^k^n l^fc^n almost everywhere.) (d) Let E(/) be the measurable set of points x e X such that /(*) + (U-f)(x) + - + (t/"-1 /)(*) > 0 for some integer n > 0. Show that > 0 ~- f JE(/) (E. Hopf's maximal ergodic theorem). (For each integer n ^> 1, let En(/) denote the set of x e X such that /(*) + ((//)(*) + - + (£/* /)W > o